TR23-182 Authors: Prahladh Harsha, Mrinal Kumar, Ramprasad Saptharishi, Madhu Sudan

Publication: 21st November 2023 19:57

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We prove that the most natural low-degree test for polynomials over finite fields is ``robust'' in the high-error regime for linear-sized fields. Specifically we consider the ``local'' agreement of a function $f\colon \mathbb{F}_q^m \to \mathbb{F}_q$ from the space of degree-$d$ polynomials, i.e., the expected agreement of the function from univariate degree-$d$ polynomials over a randomly chosen line in $\mathbb{F}_q^m$, and prove that if this local agreement is $\epsilon \geq \Omega((\frac{d}{q})^\tau))$ for some fixed $\tau > 0$, then there is a global degree-$d$ polynomial $Q\colon\mathbb{F}_q^m \to \mathbb{F}_q$ with agreement nearly $\epsilon$ with $f$. This settles a long-standing open question in the area of low-degree testing, yielding an $O(d)$-query robust test in the ``high-error'' regime (i.e., when $\epsilon \frac{1}{2}$ (Polishchuk & Spielman, STOC 1994), or $q = \Omega(d^4)$ (Arora & Sudan, Combinatorica 2003), or needed to measure local distance on $2$-dimensional ``planes'' rather than one-dimensional lines leading to $\Omega(d^2)$-query complexity (Raz & Safra, STOC 1997).

Our analysis follows the spirit of most previous analyses in first analyzing the low-variable case ($m = O(1)$) and then ``bootstrapping'' to general multivariate settings. Our main technical novelty is a new analysis in the bivariate setting that exploits a previously known connection between multivariate factorization and finding (or testing) low-degree polynomials, in a non ``black-box'' manner. This connection was used roughly in a black-box manner in the work of Arora \& Sudan --- and we show that opening up this black box and making some delicate choices in the analysis leads to our essentially optimal analysis. A second contribution is a bootstrapping analysis which manages to lift analyses for $m=2$ directly to analyses for general $m$, where previous works needed to work with $m = 3$ or $m = 4$ --- arguably this bootstrapping is significantly simpler than those in prior works.